
A new \(k\) -way \(p\) -spectral clustering method is proposed. The problem of finding \(k\)  eigenpairs of the graph \(p\) -Laplacian is recast as a Riemannian optimization problem on a Grassmann manifold, thus preserving the orthogonality between the resulting eigenvectors. We therefore succeed in reformulating a constrained minimization problem into an unconstrained one on a manifold, a prevailing trend in optimization [1]}.

An analysis of existing work in \(k\) -way \(p\) -spectral clustering is provided. Deficiencies in former formulations of the problem are identified and a correction is proposed to overcome them.

Special emphasis is put in the minimization of the nonlinear
objective function. This is achieved by means of a Grassmannian Newton method, with a truncated conjugate gradient algorithm for the linear intermediate steps [2]}. The fact that the problem lies on a manifold of significantly lower dimension than that of the Euclidean space results in lower complexity of the optimization algorithm and better numerical properties than the current state of the art in spectral clustering methods [3]}.

Our approach to estimate multiple eigenvectors of the graph \(p\) -Laplacian \(\Delta _p\)  on a manifold with intrinsic orthogonality constraints leverages the benefits from both direct \(k\) -way clustering methods and the advantages of reformulating spectral clustering in the \(p\) -norm.

A comprehensive study on both artificial datasets and machine learning datasets
demonstrates the benefit of our approach. The broad applicability and the sustainability is validated by using a wide set of 36 graphs emerging from machine learning datasets. We demonstrate that \(k\) -way \(p\) -spectral clustering performs consistently better compared to standard spectral clustering techniques.

